Skip to main content
SpringerLink
Log in

Space-Time Percolation

  • Chapter
In and Out of Equilibrium 2

Part of the book series: Progress in Probability ((PRPR,volume 60))

Abstract

The one-dimensional contact model for the spread of disease may be viewed as a directed percolation model on ℤ×ℝ in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The corresponding process when the time-axis is unoriented is an undirected percolation model to which now standard techniques may be applied. One may construct in similar vein a random-cluster model on ℤ×ℝ, with associated continuum Ising and Potts models. These models are of independent interest, in addition to providing a path-integral representation of the quantum Ising model with transverse field. This representation may be used to obtain a bound on the entanglement of a finite set of spins in the quantum Ising model on ℤ, where this entanglement is measured via the entropy of the reduced density matrix. The mean-field version of the quantum Ising model gives rise to a random-cluster model on K n ×ℝ, thereby extending the Erdős-Rényi random graph on the complete graph K n .

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Aizenman and P. Jung. On the critical behavior at the lower phase transition of the contact process, 2006. arxiv:math.PR/0603227.

    Google Scholar 

  2. M. Aizenman, A. Klein, and C.M. Newman. Percolation methods for disordered quantum Ising models. In R. Kotecký, editor, Phase Transitions: Mathematics, Physics. Biology,..., pages 129–137. World Scientific, Singapore, 1992.

    Google Scholar 

  3. M. Aizenman and B. Nachergaele. Geometric aspects of quantum spin states. Communications in Mathematical Physics, 164:17–63, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  4. K. Alexander. On weak mixing in lattice models. Probability Theory and Related Fields, 110:441–471, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  5. K. Alexander. Mixing properties and exponential decay for lattice systems in finite volumes. Annals of Probability, 32:441–487, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  6. L. Amico, R. Fazio, A. Osterloh, and V. Vedral. Entanglement in many-body systems. arxiv:quant-ph/0703044.

    Google Scholar 

  7. E. Andjel. Survival of multidimensional contact process in random environments. Boletim da Sociedade Brasileira de Matemática, 23:109–119, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  8. I. Benjamini and O. Schramm. Conformal invariance of Voronoi percolation. Communications in Mathematical Physics, 197:75–107, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  9. C.E. Bezuidenhout and G.R. Grimmett. The critical contact process dies out. Annals of Probability, 18: 1462–1482, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  10. C.E. Bezuidenhout and G.R. Grimmett. Exponential decay for subcritical contact and percolation processes. Annals of Probability, 19:984–1009, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Björnberg, 2007. In preparation.

    Google Scholar 

  12. B. Bollobás. Random Graphs, Cambridge University Press, 2nd edition, 2001.

    Google Scholar 

  13. B. Bollobás, G.R. Grimmett, and S. Janson. The random-cluster model on the complete graph. Probability Theory and Related Fields, 104:283–317, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  14. B. Bollobás, S. Janson, and O. Riordan. The phase transition in inhomogeneous random graphs. Random Structures and Algorithms, 2007. arxiv:math.PR./0504589.

    Google Scholar 

  15. B. Bollobás and O. Riordan. The critical probability for random Voronoi percolation in the plane is 1/2. Probability Theory and Related Fields, 136:417–468, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  16. B. Bollobás and O. Riordan. Percolation. Cambridge University Press, 2006.

    Google Scholar 

  17. M. Campanino, A. Klein, and J.F. Perez. Localization in the ground state of the Ising model with a random transverse field. Communications in Mathematical Physics, 135:499–515, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Cerf. The Wulff Crystal in Ising and Percolation Models, Springer, Berlin, 2006.

    MATH  Google Scholar 

  19. L.-C. Chen and A. Sakai, Critical behavior and the limit distribution for long-range oriented percolation, I. 2007.

    Google Scholar 

  20. L.-C. Chen and N.-R. Shieh. Critical behavior for an oriented percolation with longrange interactions in dimension d>2. Taiwanese Journal of Mathematics, 10:1345–1378, 2006.

    MATH  MathSciNet  Google Scholar 

  21. T. Dorlas. Probabilistic derivation of a noncommutative version of Varadhan’s theorem. 2002.

    Google Scholar 

  22. S.N. Ethier and T.G. Kurtz. Markov Processes, Wiley, New York, 1986.

    MATH  Google Scholar 

  23. M. Fannes, H. Spohn, and A. Verbeure. Equilibrium states for mean field models. Journal of Mathematical Physics, 21:355–358, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  24. D.S. Fisher. Critical behavior of random transverse-field Ising spin chains. Physical Review B, 51:6411–6461, 1995.

    Article  Google Scholar 

  25. H.-O. Georgii and T. Küneth. Stochastic comparison of point random fields. Journal of Applied Probability, 34:868–881, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  26. P. Grassberger and A. de la Torre. Reggeon field theory (Schögl’s first model) on a lattice: Monte Carlo calculations of critical behaviour. Annals of Physics, 122:373–396, 1979.

    Article  Google Scholar 

  27. G.R. Grimmett. Percolation. Springer, Berlin, 1999.

    MATH  Google Scholar 

  28. G.R. Grimmett. The Random-Cluster Model. Springer, Berlin, 2006.

    MATH  Google Scholar 

  29. G.R. Grimmett and S. Janson. Branching processes, and random-cluster measures on trees. Journal of the European Mathematical Society, 7:253–281, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  30. G.R. Grimmett, T.J. Osborne, and P.F. Scudo. Entanglement in the quantum Ising model. Journal of Satistical Physics, 131:305–339, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  31. O. Häggstrom. The random-cluster model on a homogeneous tree. Probability Theory and Related Fields, 104:231–253, 1996.

    Article  MathSciNet  Google Scholar 

  32. D. Ioffe and A. Levit. Long range order and giant components of quantum random graphs. Markov Processes and Related Fields, 13:469–492, 2007.

    MATH  MathSciNet  Google Scholar 

  33. S. Janson. On a random graph related to quantum theory. Combinatorics, Probability and Computing, 16:757–766, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  34. S. Janson, T. Łuczak, and A. Ruciński. Random Graphs. John Wiley, New York, 2000.

    MATH  Google Scholar 

  35. A. Klein. Extinction of contact and percolation processes in a random environment. Annals of Probability, 22:1227–1251, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  36. A. Klein. Multiscale analysis in disordered systems: percolation and contact process in random environment. In G.R. Grimmett, editor, Disorder in Physical Systems, pages 139–152. Kluwer, Dordrecht, 1994.

    Google Scholar 

  37. G. Lawler. Conformally Invariant Processes in the Plane. American Mathematical Society, 2005.

    Google Scholar 

  38. T.M. Liggett, Interacting Particle Systems. Springer, Berlin, 1985.

    MATH  Google Scholar 

  39. T.M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin, 1999.

    MATH  Google Scholar 

  40. M. Luczak and T. Łuczak. The phase transition in the cluster-scaled model of a random graph. Random Structures and Algorithms, 28:215–246, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  41. R. Meester and R. Roy. Continuum Percolation. Cambridge University Press, 1996.

    Google Scholar 

  42. C.M. Newman and S. Volchan. Persistent survival of one-dimensional contact processes in random environments. Annals of Probability, 24:411–421, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  43. M.D. Penrose. Random Geometric Graphs. Oxford University Press, 2003.

    Google Scholar 

  44. C.J. Preston. Spatial birth-and-death processes. Bulletin of the International Statistical Institute, 46:371–391, 405–408, 1975.

    MathSciNet  Google Scholar 

  45. A. Sakai and R. van der Hofstad. Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions. Probability Theory and Related Fields, 132:438–470, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  46. O. Schramm. Conformally invariant scaling limits: an overview and a collection of open problems. In Proceedings of the International Congress of Mathematicians, Madrid, 2006, volume I, pages 513–544. European Mathematical Society, Zürich, 2007.

    Google Scholar 

  47. G. Slade. The Lace Expansion and its Applications. Springer, Berlin 2006.

    Google Scholar 

  48. S. Smirnov. Towards conformal invariance of 2D lattice models. In Proceedings of the International Congress of Mathematicians, Madrid, 2006, volume II, pages 1421–1452. European Mathematical Society, Zürich, 2007.

    Google Scholar 

  49. W. Werner. Random planar curves and Schramm-Loewner evolutions. In J. Picard, editor, Ecole ďEté de Probabilités de Saint Flour XXXII-2002, pages 107–195. Springer, Berlin, 2004.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Geoffrey R. Grimmett

Authors
  1. Geoffrey R. Grimmett
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina, 110 Jardim Botanico, CEP 22460-320, Rio de Janeiro, RJ, Brasil

    Vladas Sidoravicius

  2. Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud, 150 Urca, 22290-180, Rio de Janeiro, RJ, Brasil

    Maria Eulália Vares

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Grimmett, G.R. (2008). Space-Time Percolation. In: Sidoravicius, V., Vares, M.E. (eds) In and Out of Equilibrium 2. Progress in Probability, vol 60. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8786-0_15

Download citation

Publish with us

Policies and ethics

Search

Navigation